Absolute Logical Truth Has No Foundations But Random Occurence

[vamvam]

Truth is a property of something and as such is relative to this thing. Absolute truth would be then property of nothing. Absolute truth depends on nothing and describes nothing.

[puto]

What?

[Eodnhoj7]

"There is no absolute truth" is false under its own context.

What is this context?

"There is no absolute truth" is not absolute otherwise it contradicts itself as being absolute, hence it relative, and as such is false under some contexts.

[Eodnhoj7]

If all contexts are completely different

Expressed in the context of first-order logic, the following sentence s would be true in every context M:

∃s∀M (M⊨s)

So, the sentence s would be an absolute truth.

However, it is not possible to quantify in first-order logic over every context M. The term "all contexts" (∀M) is therefore not supported.

This is a problem similar to the problem of trying to deal with "the set of all sets" in ZFC set theory. Such set would be in violation of the axiom of foundation and the axiom of separation.

You would need a context in which the notion of "all contexts" is definable. Such context may not exist.

I will address your foundation, ie the first sentence...or at least a part of it.

First order logic is a context of interpretation.

There is no law as to what or how to choose a context by which to interpret.

This absence of law necessitates a spontaneous occurence of absolute truth. Spontaneity is a soft word for randomness.

[godelian]

"There is no absolute truth" is not absolute otherwise it contradicts itself as being absolute, hence it relative, and as such is false under some contexts.

Natural language is too liberal for the purpose of investigating corner cases in logic.

[Eodnhoj7]

"There is no absolute truth" is not absolute otherwise it contradicts itself as being absolute, hence it relative, and as such is false under some contexts.

Natural language is too liberal for the purpose of investigating corner cases in logic.

If that is the case than how can what you just said about natural language make any real sense?

[godelian]

"There is no absolute truth" is not absolute otherwise it contradicts itself as being absolute, hence it relative, and as such is false under some contexts.

Natural language is too liberal for the purpose of investigating corner cases in logic.

If that is the case than how can what you just said about natural language make any real sense?

You can express this limitation of natural language in natural language itself.

You can also express limitations of first-order logic (FOL), such as Godel's incompleteness theorem, in itself, on the condition that FOL implements enough arithmetic to that effect.

DeepSeek: You can express the limitation of natural language in natural language itself.

Indeed, the limitation of natural language can be expressed within natural language itself, highlighting its inherent constraints even as it describes them. Here's one way to articulate it:

*"Natural language can point to its own ambiguities, vagueness, and incompleteness, but it cannot fully escape them. In naming its limitations, it must still rely on the very medium it critiques—like a net trying to grasp its own weave."*

This self-referential paradox mirrors broader philosophical and linguistic issues (e.g., the "liar paradox," Wittgenstein's "ladder," or Gödel's incompleteness theorems). Language can *describe* its flaws but not *resolve* them without stepping outside itself—which, for human communication, is impossible.

There's no way, however, to properly formalize and very precisely pinpoint in the context of natural language what the limitation is, such as Godel's incompleteness theorem in the context of first-order arithmetic.

So, in the case of "there is no absolute truth", there is no way to properly formalize or pinpoint precisely what it means in natural language. Natural language simply does not define the terms involved well enough.

[Eodnhoj7]

Natural language is too liberal for the purpose of investigating corner cases in logic.

If that is the case than how can what you just said about natural language make any real sense?

You can express this limitation of natural language in natural language itself.

You can also express limitations of first-order logic (FOL), such as Godel's incompleteness theorem, in itself, on the condition that FOL implements enough arithmetic to that effect.

DeepSeek: You can express the limitation of natural language in natural language itself.

Indeed, the limitation of natural language can be expressed within natural language itself, highlighting its inherent constraints even as it describes them. Here's one way to articulate it:

*"Natural language can point to its own ambiguities, vagueness, and incompleteness, but it cannot fully escape them. In naming its limitations, it must still rely on the very medium it critiques—like a net trying to grasp its own weave."*

This self-referential paradox mirrors broader philosophical and linguistic issues (e.g., the "liar paradox," Wittgenstein's "ladder," or Gödel's incompleteness theorems). Language can *describe* its flaws but not *resolve* them without stepping outside itself—which, for human communication, is impossible.

There's no way, however, to properly formalize and very precisely pinpoint in the context of natural language what the limitation is, such as Godel's incompleteness theorem in the context of first-order arithmetic.

So, in the case of "there is no absolute truth", there is no way to properly formalize or pinpoint precisely what it means in natural language. Natural language simply does not define the terms involved well enough.

You can express limitations and you cannot. Something as simple as "one" or "1" appears distinct in one respect but in another is either too abstract or applies to too many things as to alleviate any sense of distinct meaning.

To be real about the word "vague" it can be asked "what does it mean?". See my point?

Yes as GPT pointed out it ends in paradox....which is what I implied with the question: If language is vague than in saying "language is vague" the meaning of language being vague becomes vague and "vagueness" on one half becomes a fruitless assertion. In other words to say "language is vague" is vague.

There is no way to properly formalize meaning in mathematics on the other hand as the simple equation of "1+1=2" can be applied to infinite things and as such loses its distinctness. A number is merely a quantification of limits and quantification can be applied to any limit or number of limits.

[godelian]

There is no way to properly formalize meaning in mathematics on the other hand as the simple equation of "1+1=2" can be applied to infinite things and as such loses its distinctness. A number is merely a quantification of limits and quantification can be applied to any limit or number of limits.

No, disagreed.

In the Platonic ontology, "1+1=2" is a truth about Platonic abstractions that exist in their own abstract Platonic universe. This has nothing to do with "applied to infinite things". They exist in and of themselves. It is not because downstream disciplines such as science or engineering "apply" these things that we see them as such in mathematics. Nothing is ever applied in mathematics. If these things is are ever "applied", then it is not math but some other downstream discipline that is none of our business.

An alternative way of looking at things, formalism, is to view these expressions, such as "1+1=2" as meaningless symbol streams that are obtained by means of meaningless string manipulation. If you correctly manipulate a sequence of meaningless symbols, then you will obtain some other meaningless symbols as a result. "Wrong" means that you did not follow the rules.

So, in mathematics, a number is never a quantification of sorts. It is an abstract object in a separate universe or it is a meaningless symbol for which there exist manipulation rules.

In connection with the physical universe, these things essentially mean nothing. Science and engineering may assign some semantics to these things within their own frameworks, but mathematics refuses to take any responsibility in that regard or even to acknowledge any such external semantics.

[Eodnhoj7]

There is no way to properly formalize meaning in mathematics on the other hand as the simple equation of "1+1=2" can be applied to infinite things and as such loses its distinctness. A number is merely a quantification of limits and quantification can be applied to any limit or number of limits.

No, disagreed.

In the Platonic ontology, "1+1=2" is a truth about Platonic abstractions that exist in their own abstract Platonic universe. This has nothing to do with "applied to infinite things". They exist in and of themselves. It is not because downstream disciplines such as science or engineering "apply" these things that we see them as such in mathematics. Nothing is ever applied in mathematics. If these things is are ever "applied", then it is not math but some other downstream discipline that is none of our business.

An alternative way of looking at things, formalism, is to view these expressions, such as "1+1=2" as meaningless symbol streams that are obtained by means of meaningless string manipulation. If you correctly manipulate a sequence of meaningless symbols, then you will obtain some other meaningless symbols as a result. "Wrong" means that you did not follow the rules.

So, in mathematics, a number is never a quantification of sorts. It is an abstract object in a separate universe or it is a meaningless symbol for which there exist manipulation rules.

In connection with the physical universe, these things essentially mean nothing. Science and engineering may assign some semantics to these things within their own frameworks, but mathematics refuses to take any responsibility in that regard or even to acknowledge any such external semantics.

If mathematics is a universal truth than you would not need natural language to argue the above.

Now:

If "1+1=2" exists in and of itself you could not possibly know that as you observing it as existing in and of itself ceases when you observe it as your observation makes it relational to you.

If 1+1=2 exists in and of itself you could not prove it because to prove it would make it conditional and cease your point.

[godelian]

If mathematics is a universal truth than you would not need natural language to argue the above.

Mathematics investigates the truth about the abstract Platonic universe only. It is not an attempt at more universality than that. Mathematics is not about the physical universe. It does not express any truth about the physical universe.

If 1+1=2 exists in and of itself you could not prove it because to prove it would make it conditional and cease your point.

It is conditional on accepting a set of basic beliefs such as Peano Arithmetic theory (PA).

PA ⊢ 1+1=2

If you do not accept something akin to PA, then you cannot prove the truth of the expression.

By the way:

GF(2) ⊢ 1+1=0

Just to point out that the expression "1+1=2" is not universally true. It is true only in a particular context.

Concerning "If 1+1=2 exists in and of itself", it is 1 and 2 that are basic Platonic objects in the standard model of arithmetic. The expression is an element of Th(ℕ), i.e. "true arithmetic":

https://en.m.wikipedia.org/wiki/True_arithmetic

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.[1] This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

So, indeed, "1+1=2" exists in and of itself as an abstract Platonic object that is also an element of Th(ℕ).

[Eodnhoj7]

If mathematics is a universal truth than you would not need natural language to argue the above.

Mathematics investigates the truth about the abstract Platonic universe only. It is not an attempt at more universality than that. Mathematics is not about the physical universe. It does not express any truth about the physical universe.

If 1+1=2 exists in and of itself you could not prove it because to prove it would make it conditional and cease your point.

It is conditional on accepting a set of basic beliefs such as Peano Arithmetic theory (PA).

PA ⊢ 1+1=2

If you do not accept something akin to PA, then you cannot prove the truth of the expression.

By the way:

GF(2) ⊢ 1+1=0

Just to point out that the expression "1+1=2" is not universally true. It is true only in a particular context.

Concerning "If 1+1=2 exists in and of itself", it is 1 and 2 that are basic Platonic objects in the standard model of arithmetic. The expression is an element of Th(ℕ), i.e. "true arithmetic":

https://en.m.wikipedia.org/wiki/True_arithmetic

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.[1] This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

So, indeed, "1+1=2" exists in and of itself as an abstract Platonic object that is also an element of Th(ℕ).

1. If the platonic universe is purely abstract than you are strictly speaking about your own thoughts as I nor anyone else can see your thoughts considering that is what abstractions are: thoughts. Please point to this platonic universe.

2. I see.... so mathematical truth is conditioned on beliefs...I am not seeing anything mathematical in that. Could you please reduce that to an equation?

[godelian]

1. If the platonic universe is purely abstract than you are strictly speaking about your own thoughts as I nor anyone else can see your thoughts considering that is what abstractions are: thoughts. Please point to this platonic universe.

Commonly held axioms lead to commonly and objectively provable theorems.

2. I see.... so mathematical truth is conditioned on beliefs...I am not seeing anything mathematical in that.

That is almost surely because you do not understand the axiomatic nature of mathematics. In general, we can state that this problem tends to occur when you don't know what you are talking about. So, you may think that you understand mathematics, but in fact, you absolutely don't.

This is a very common phenomenon.

[Eodnhoj7]

1. If the platonic universe is purely abstract than you are strictly speaking about your own thoughts as I nor anyone else can see your thoughts considering that is what abstractions are: thoughts. Please point to this platonic universe.

Commonly held axioms lead to commonly and objectively provable theorems.

2. I see.... so mathematical truth is conditioned on beliefs...I am not seeing anything mathematical in that.

That is almost surely because you do not understand the axiomatic nature of mathematics. In general, we can state that this problem tends to occur when you don't know what you are talking about. So, you may think that you understand mathematics, but in fact, you absolutely don't.

This is a very common phenomenon.

Commonality is merely another term for generally, generally is another way of saying probably. It appears mathematical truth has political elements of a democracy.

I am asking you to resort to arguing your case using strictly mathematics if natural language is ambiguous.

[godelian]

I am asking you to resort to arguing your case using strictly mathematics if natural language is ambiguous.

Philosophy is not even meant to be provable.

Furthermore, there is nothing provable in the philosophy of mathematics. The philosophy of mathematics is not axiomatic and is not a subdivision of mathematics.

You are confusing the philosophy of mathematics with metamathematics, which is indeed axiomatic and a subdivision of mathematics.